Cars run without a thermostat will still overheat. Besides opening and closing to regulate the engine temperature, it also regulates how fast the water flows through the radiator. If the hole is too small, the engine will over heat but the coolant in the radiator will be below operating temperature. This will also happen if the thermostat freezes closed. You can tell when this happens, the coolant in the upper radiator hose will make a chugging noise.
If the hole in the thermostat is too big or if the thermostat is missing, the coolant will flow through the radiator too fast to cool down to the proper temperature. Your car will take longer than normal to get to the preferred operation temp then will slowly overheat. This type of overheating can cause the most engine damage. The whole system is hot and the radiator cap can’t release the pressure fast enough. Head gaskets will blow, blocks and heads will warp and radiators will split at the seams.
If you are ever stuck somewhere with a bad thermostat, a good temporary fix it to bend open or remove the center blocking plate. In the 106 years of automotive history and with over 6,000,000,000 (yes, 6 billion) automobiles built in that time, I would bet that not once has a car failed to start on a 50 degree morning because of a bad thermostat.